fundamental extended objects for chern–simons supergravity

1 June 2000

Ž .Physics Letters B 482 2000 222–232

Fundamental extended objects for Chern–Simons supergravity q

Pablo Mora 1, Hitoshi NishinoDepartment of Physics UniÕersity of Maryland at College Park, College Park, MD 20742-4111, USA

Received 20 January 2000; received in revised form 17 April 2000; accepted 20 April 2000Editor: M. Cvetic

Abstract

We propose a class of models in which extended objects are introduced in Chern–Simons supergravity in such a way thatthose objects appear on the same footing as the target space. This is motivated by the idea that branes are already firstquantized object, so that it is desirable to have a formalism that treats branes and their target space in a similar fashion.Accordingly, our models describe interacting branes, as gauge systems for supergroups. We also consider the case in whichthose objects have boundaries, and discuss possible links to superstring theory andror M-theory, by studying the fermionick-symmetry of the action. q 2000 Elsevier Science B.V. All rights reserved.

1. Introduction

Gauge supergravity theories, based on lagrangiansof the Chern–Simons form involving gauge fieldsfor a gauge supergroup containing the super Poincaregroup of the appropriate dimension, have recentlyattracted a great deal of attention. Since the construc-

Ž . w xtion of 2q1 -dimensional theories 1–3 and itsw xquantization 2 these theories have been extended to

w xhigher dimensions by Chamseddine 4 and furtherw xdeveloped by Banados, Troncoso and Zanelli 5,6 .˜

w x w xAlso Green 7 , or Moore and Seiberg 8 gavepossible links between Chern–Simons theory for a

q This work is supported in part by NSF grant a PHY-93-41926.Ž .E-mail addresses: [email protected] P. Mora ,

Ž [email protected] H. Nishino .1 ´Also at Instituto de Fisica, Facultad de Ciencias, Igua 4225,´

Montevideo, Uruguay.

w xsupergroup and superstring theory, while Horava 9ˇw xsuggested that M-theory 10–12 may actually be a

field theory of this class.In this Letter we will suggest a natural way to

introduce fundamental extended objects in Chern–Simons supergravities by embedding lower dimen-sional lagrangians for Chern–Simons supergravity inhigher dimensions. We may also think of this modelas a set of Chern–Simons supersymmetric branes2

embedded into a background described also by aŽChern–Simons action and can be considered itself a

Chern–Simons brane, even though it is not embed-.ded in a larger manifold providing in effect a

‘brane-target space democracy’. It is important toremark, however, that supersymmetry is introducedhere as a gauge symmetry, in the sense that the super

ŽPoincare group or any of it extensions, like the´

2 Chern–Simons bosonic branes in a somewhat different frame-w xwork have been considered in Ref. 13 .

0370-2693r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 00 00535-9

( )P. Mora, H. NishinorPhysics Letters B 482 2000 222–232 223

Ž . .super conformal or super anti de Sitter groups arerealized as parts of a gauge group. That is to becontrasted with the standard approach of gettingspacetime supersymmetry by defining the extendedobjects as embeddings in flat or curved superspace.

A heuristic motivation for the kind of modelspresented here is that if there is some underlyingpregeometric theory in which the de Rham complexget physical content through some kind of antighostfield variables, we must expect the effective action tobe a sum of pieces of diverse dimensions of the formconsidered here. We believe that this simple observa-tion is the key to understand the existence of ex-tended objects in a fundamental theory of nature.

We also consider the case in which the embeddedsupersymmetric branes have boundaries. In such acase, we need to add boundary terms in order topreserve gauge invariance in a way resembling the

w xaction of Dixon et al. 14 for extended objectscoupled to gauge fields except in two very importantrespects: First, we implement supersymmetry as a

Ž .part of the gauge super group. Second, we have aŽ .bulk s with nontrivial degrees of freedom, which

can be ignored only in the case that the gaugepotential for the supergroup is pure-gauge. We willshow that in this case and for a particular choice ofthe gauge supergroup one can establish a link with

Ž . w xten-dimensional 10D superstring theory 15 .

2. Action for extended objects

Ž .We consider a super group G with generatorsT I, the gauge potential 1-form AsA IT dx m andm I

the curvature 2-form FsdAqA2, defined on amanifold M Dq 1 of even dimension Dq1. If we

Ž .have an invariant polynomial P F defined as theformal sum

Ž .Dq1 r2nP F s a STr F , 1Ž . Ž . Ž .Ý n

ns1

here STr T I1 . . . T In sg I1 PPP In stands for an invari-Ž .ant symmetric trace on the algebra of G, we definethe action for our system by

Dy10Ss a I . 2Ž .Ý Hd dq1

dq1Vd even

0 Ž .Here the I F, A are the Chern–Simons formsdq1Ž .associated with P F , so locally

dq2Dy1

2P F s a STr FŽ . Ý d ž /d even

Dy1 d0s a dI F , A , ns q1 ,Ž .Ý d dq1 ž /2d even

3Ž .

and M Dq 1 has V D as its boundary, while theV dq1’s are submanifolds of V D of odd dimension.The dynamical variables are the gauge potentials Aand the embedding coordinates of the submanifolds

m Ž i . iX x , where the x with is1, . . . , dq1 areŽd . Žd . Žd .local coordinates in V dq1. Notice that all thesemanifolds are supposed to be non compact at least inwhat would be the ‘time direction’. As observed in

w xRef. 6 , each of the dimensionless coefficients ad

can consistently take only a discrete set of values asa result of the requirement that the action must beindependent of the way in which the manifoldsV dq1 are extended into manifolds M dq2 includedinto M Dq 1 such that EV dq1 'M dq2. 3

It seems to us that even though any symmetrictrace is allowed at this level, if we are looking for aeffective theory corresponding to some underlyingpregeometric theory then Chern characters may befavored on the grounds of their additivity under a

Ž .Stieffel direct sum of bundles, which would corre-spond to overlapping ‘brane elements’ at a point.However we will keep our considerations as generalas possible in what follows.

Suppose now that we allow for the possibility thatthe extended objects with world-volume V dq1 haveboundaries Sd 'EV dq1. In that case the action isnot gauge invariant anymore, because of a boundaryterm in the gauge variation d of the Chern–SimonsG

w xactions 16,17 . Locally

d I 0 F , A sdI 1 F , A ,l ,Ž . Ž .G dq1 d

where we are using the notation conventions of Ref.w x Ž14 . One way to cancel this variation actually also

3 Ž . w i FrŽ2p . xFor instance P F s STr e would work.

( )P. Mora, H. NishinorPhysics Letters B 482 2000 222–232224

.the global variation is to add to the action boundaryw x4 Žterms of a form analogous to the one in 14 see

w x.also Ref. 18 so that the complete action now reads

Dy10Ss a IÝ Hd dq1½ dq1Vd even

1dq d z yg(H Ž . Ž .d dd2 S

= i jg STr J J y dy2Ž .Ž .Ž .i jŽ .d

y C A , K yb , 4Ž . Ž .Ž .H d d 5dS

where the z ’s are local coordinates in Sd and

E X m E y i

I I IJ sA yK ,i m ii iEz Ez

or JsAyK as 1-form 5Ž . Ž .where K is the left-invariant Maurer–Cartan form in

Ž . ithe super group manifold G, the y ’s are coordi-Ž . Ž .nates on that super group manifold, while K y si

y1Ž . Ž . i 2 Žg y E g y rE y satisfies dKqK s0 the differ-d .entials are taken in S for y-dependent quantities ,

0 Ž .and b is a d-form such that db syI 0, K ,d d dq1Ž .while C A, K is chosen in order to preserve gauged

invariance in a way described below. The auxiliarymetrics g have purely algebraic equation of motionŽd .

that can be used to eliminate it from the action. Theb -term is a generalization of Wess–Zumino–Novi-d

Ž .kov–Witten WZNW -term. For example, the ds2Ž .case corresponds to the usual super string formula-

w xtion 15 on ds2 world-sheet with an additional 3Dbulk term with I 0. We must add to the list of our3

mŽ i . Ždynamical variables the functions X z orŽd . Žd .k Ž i .. i Ž i . i jŽ k .equivalently x z , y z and g z .Žd . Žd . Žd . Žd . Žd . Žd .

4 It is important however to point out that our action is not justw xthe action of 14 for a supergroup, as for our action there is no

Ž .background metric or Ramond–Ramond RR p-form fields, andfurthermore our branes do have a bulk with non trivial degrees offreedom and a nontrivial dynamics while the kinetic term ‘lives’

w xat the boundary. The branes of 14 would correspond to theŽboundary of ours in the pure gauge case when bulk degrees of

.freedom go away , and if we use the WZNW term from A tomimic the RR field.

Ž .Under super gauge transformations, each term ofw xour action transforms as 14

w x w xd Asdlq A ,l , dKsdlq K ,l , 6Ž .G

and accordingly J transforms covariantly

w xd Js J ,l . 7Ž .G

Also

d I 0 F , A sdI 1 F , A ,l ,Ž . Ž .G dq1 d

d b syI 1 0, K ,l , 8Ž . Ž .G d d

and we require

d C s I 1 F , A ,l y I 1 0, K ,l q exact form ,Ž . Ž . Ž .G d d d

9Ž .w xwhich is satisfied for 14

10 0C sk I F , A s dt l I F , A , 10Ž . Ž . Ž .Hd 01 dq1 t t t dq1 t t0

where we identify

A'A , K'A , 11Ž .1 0

for

A s tA q 1y t A ,Ž .t 1 0

22F sdA qA s tFq t ty1 A yA . 12Ž . Ž . Ž .t t t 1 0

Ž .For ds2 for instance, C s STr AK . The l in2 tŽ .Eq. 10 is defined to act on arbitrary polynomials

l A s0 , l F sA yA , 13Ž .t t t t 1 0

with the convention that l is defined to act as ant

antiderivation. Then the Cartan homotopy operator isw xdefined by 17

1k PP F , A s d t l PP F , A , 14Ž . Ž . Ž .H01 t t t t t

0

where d t is for the t-integration, to be distinguishedw xfrom other one-forms such as dA 17 . By integrat-t

ing

El dqdl PP F , A s PP F , A 15Ž . Ž . Ž . Ž .t t t t t tE t

over t from 0 to 1, we get the Cartan homotopyw xformula 17

k dqdk PP F , A sPP F , AŽ . Ž . Ž .01 01 t t 1 1

yPP F , A . 16Ž . Ž .0 0

Ž .Putting all this together we see that the action 4 isindeed gauge invariant.


Ž .A more compact form of the action 4 can begiven by considering the Cartan homotopy formula

Ž . 0 Ž . Ž .for PP F , A s I F , A , then the action 4 ist t dq1 t t

rewritten as

dq2Dy1

2Ss a k STr FÝ Hd 01 tž /½ dq1Vd even

1dq d z yg(H Žd . Žd .

d2 S

= i jg STr J J y dy2 . 17Ž . Ž .Ž .i jŽ .d 50 Ž .dq 1Here we used also the relationship H I 0, KV dq1

syH d b . This form of the action makes its gaugeS d

invariance more manifest, because k is a gauge01

invariant operator. We can generalize our action onestep further, and treat A sK and A sA in a0 1

symmetric fashion by letting A be an arbitrary0

gauge field and not just pure-gauge, then the actionŽ . Ž .4 is just a particular case of the action 17 . Theprice we have paid is doubling the number of bulkdegrees of freedom.5

Ž 2 .It is worthwhile to note that the STr J in theŽ .‘kinetic’ term in the action 4 is gauge invariant by

itself, while the bulk and the other boundary termstransform into each other under gauge transforma-tions. That would in principle mean that the coeffi-cient of that term can be chosen independently.However, as will be seen in the next section, thek-symmetric Green–Schwarz string action obtainedas a certain limit fixes that coefficient, because thek-symmetry mixes the kinetic and WZNW-term ofthe action. In the standard formulation of the het-erotic string, we have the choice between a

5 ŽThe Cauchy problem as well as causality issues, considering. Ž .the absence of a background metric for the action 17 seems

formidable, though probably not much worse than the sameproblems for standard Chern–Simons supergravities. In the casethat the manifold in which all the branes are embedded hasboundaries, we would need boundary conditions for the gaugefields, as well as specifying whether the extended objects haveboundaries on that boundary or not, in order to have a well posedCauchy problem.

w x w x‘fermionic’ 19 and a ‘bosonic’ 20 formulation,depending on whether the gauge group acts onworld-sheet fermions or on bosonic group manifoldcoordinates. Our action is analogous to the latter, butwe might think of an alternative fermionic formula-tion in which the quantum anomaly of a quadratickinetic boundary term cancels out the boundary gaugevariation of the Chern–Simons bulk term, and theWZNW-term is a quantum effect, so that all relativecoefficients are determined.

The k-symmetry should in principle rule out theinfinite number of generally covariant and gaugeinvariant terms that we can build out of the gauge

Ž .covariant objects F and J and their pull-backs andthe gauge invariant g ’s. However, some alternativeforms of the kinetic term may also yield k-symmet-ric actions, for instance Born–Infeld-like kineticterms as

d Ž . Ž .� 4d z STr y sdet J J q F q F 18aŽ .(H i j i jŽd . i j 0 1dS

or

d Ž . Ž .d z STr y sdet STr J J q F q F� 4Ž .(H i j i jŽd . i j 0 1dS

18bŽ .

where the superdeterminant is taken in the curvedindices i, j of the pull-backs on Sd while the super-traces are taken on the group indices. Notice thatbecause supersymmetry is realized as a part of thegauge group for our models the kinetic terms givenbefore are at once a gauge non-abelian and fermionic

Ž .supersymmetric as well as generally covariant gen-eralization of the bosonic abelian Born–Infeld ac-tion. Out of these possible alternative kinetic terms,the second one is the one that corresponds to a morestraightforward generalization of the metric-eliminated version of our original one. The aboveactions with Born–Infeld-like kinetic terms wouldprovide us a framework where fundamental stringsand Dp-branes can be treated in an equivalent fash-ion. Duality arguments would likely help in fixingthe precise form of the kinetic term. For a review ofthe literature on D-branes and Born–Infeld terms seew x Ž21 and references therein. Cf. A related recent

w x.work in 22


Ž . Ž .Notice that our actions of Eqs. 4 and 17describe a system of any number of interacting braneswith or without boundaries which are charged withrespect to the gauge fields of the model. The currentscarried by the branes can be computed by taking thefunctional derivatives of different pieces of the ac-tion with respect to the gauge potentials.

( < )3. From OSp 32 1 to Type IIA Green–Schwarzsuperstring

We next give some intermediate steps to obtainthe action for 10D Type IIA Green–Schwarz super-string from a Maurer–Cartan form for the M-theory

Ž < . w xgroup OSp 32 1 in 11D 9,12,23 . Subsequently, weconsider how we can understand the usual fermionic

w xk-symmetry 24 for the Green–Schwarz superstringw x15 in our formulation. We hope this provides uswith a clearer picture about the link between our

Ž . Ž .extended objects described by our action 4 or 17w xon Chern–Simons supergravity 4–6,9 with 10D

w xGreen–Schwarz superstring 15 .An earlier work towards establishing a link be-

tween Chern–Simons theory for a supergroup andw x Žsuperstring theory has been done by Green in 7 Cf.

w x.also 8 . However, several differences with ourmodel should be clarified here:

Ž .i In the last section, a Chern–Simons action fora supergroup extension of the super Poincare groupís discussed for a three dimensional base manifold,however the action does not contain boundary termswhich would render it gauge invariant withoutgauge-fixing in the boundary. Also a gauge-invariant

w xkinetic term is absent in 7 .Ž .ii In this section, the WZNW action correspond-

ing to the Chern–Simons action of Section 2 will beconsidered with a kinetic term now included, but insuch a way that the model is again not gauge invari-ant without gauge-fixing.

Ž .iii Our branes are embedded in a larger space-time and interacting with other branes, nemely, thepure gauge condition is not required for us in gen-

w xeral, as it is for Ref. 7 . That means that we canconsider generic backgrounds, and not only flat su-perspace.

Consider the group G as the M-theory groupŽ < .OSp 32 1 with generators P , Q , M and Z ,a a ab a . . . a1 5

and the symmetric trace is just the standard sym-metricized supertrace in the adjoint representation ofG. If we consider the case of ‘elementary’ funda-

Žmental objects of size one in proper coordinates as.opposite to ‘cosmic’ extended objects , living in a

large background6 then dimensional arguments showthat the n-dimensional integrals in our action scale asŽ n.OO 1rL where L is the characteristic scale of the

background. That means that the dominant terms inthe action will be those of lower dimensions that are

Ž < .non-vanishing. In the case of the group OSp 32 1the generators are traceless in the adjoint representa-

Ž .tion, so that the 0-brane particle terms are absentand the dominant contribution comes from the 2-brane. We notice that if As0, and the pure-gaugeK is of the form Ksgy1dg, then we compute

Ž y1r2 .essentially up to OO R -terms. This is equivalentto keeping only X a ,u a among the coordinatesŽ .

y i . Plugging this ansatz back into the action,Ž .restricting the boundary of the 2-brane to a 10Dboundary of the 11D base manifold, identifying thegauge parameters X a with the Ds11 coordinates,we get the kinetic term from STr J 2 in the origi-Ž .nal action, and the WZNW-term from the b -term.2

Ž < . w xThe OSp 32 1 algebra is dictated by 25,26

1 âbˆ ˆQ ,Q sa g M� 4 ˆŽ . âba b 2 abˆR1

a PPP aˆ ˆ1 5 ˆq ia g Z , 19aŽ .Ž .ˆ ab5 a PPP aˆ ˆ1 5'R

1bM ,Q sy g Q ,ˆŽ .ˆ âb a ab bˆ ˆ a2

ˆ ˆcd w c d xˆ ˆˆ ˆ ˆM , M sy4d M , 19bŽ .ˆ âb w a b xˆ ˆ

1bZ ,Q s ib g Q ,ˆŽ .a PPP a a 3 a PPP a bˆ ˆ ˆ ˆ1 5 1 5 a'R

c PPP c w c c PPP c xˆ ˆ ˆ ˆ ˆ1 5 1 1 5ˆ ˆ ˆM ,Z sy10d Z , 19cŽ .ˆ âb w a b xˆ ˆ

ˆ ˆZ ,Z ˆ â PPP a b PPP bˆ ˆ1 5 1 5

1c PPP cˆ ˆ1 5 ˆs f Z , 19dŽ .ˆ â PPP a b PPP b c PPP cˆ ˆ ˆ ˆ1 5 1 5 1 5'R

6 Notice that this is a priori a somewhat unnatural limit in ourapproach.


Ž .in the SO 1,10 manifest notation. The a’s and b’sŽ .are appropriate constants, and f ’s in 19d is a

ˆ ˆw xstructure constant for the Z,Z -commutator, whosedetail is not important here. Here all the indices andoperators with hats refer to the 11D,7 to be distin-guished from those in 10D into which we are nowgoing to perform the compactification on a torusfrom the former. The constant R denotes the radiusof such a torus. We assigned appropriate powers ofR in such a way that in the limit R™` recovers theusual super Poincare algebra in 10D, with the consis-´

ˆ ˆw x w xtent mass dimensions, namely P sq1, M â M ab Mˆ ˆˆw x w xs0, Z sq1r2, Q sq1r2. For ex-a PPP a M a Mˆ ˆ1 5 ˆw xample, the half-integer for Z sq1r2 is requiredM

ˆ Ž .by the vanishing of the Z-term in 19a for R™`.Upon the compactification for a finite R, the

algebra above becomes what is called 10D de-Sitterw xalgebra 26 , where in particular, the components of

M with the 10-th coordinate, are identified withâbˆthe momentum operator P in 10D:a

M sM ,ab abM s 20Ž .âbˆ ½ ˆ ˆM sRP syM .10 b b b 10

Ž .Thus the algebra 19 becomes the de-Sitter algebraw xin 10D 26 :

1c abQ ,Q s ia g g P qa g M� 4 Ž . Ž . aba ba b 1 11 c 2 abR

1a PPP a1 5q ia g Z , 21aŽ . Ž .ab5 a PPP a1 5'R

cd w c d xM , M sy4d M ,ab w a b x

w xM , P sq2h P , 21bŽ .ab c cw b ax

1w xP , P s ib M ,a b 1 ab2R

1bw xM ,Q sy g Q ,Ž .ab a ab ba2

1bw xP ,Q s ib g g Q , 21cŽ . Ž .a a 2 11 a baR

7 For Q’s we do not need this distinction, because all of itsMajorana components are kept in 10D upon the compactification.Relevantly, the indices a , b , PPP are Majorana indices in 10D.

1bZ ,Q s ib g Q ,ˆŽ .a PPP a a 3 a PPP a bˆ ˆ ˆ ˆ1 5 1 5 a'R

c PPP c w c c PPP c xˆ ˆ ˆ ˆ ˆ1 5 1 2 5ˆ ˆ ˆM ,Z sy10d Z , 21dŽ .ˆ âb w a b xˆ ˆ

1a PPP a w a a PPP a xˆ ˆ ˆ ˆ ˆ1 5 1 2 5ˆ ˆP ,Z sy5 d Z ,b 10 bR

1 XXˆw5xˆ ˆ ˆX X XXZ ,Z s f Z , 21eŽ .ˆ ˆ ˆ ˆ ˆw5x w5x w5xw5x w5x'R

where all other independent commutators are vanish-ˆw xing. The index 5 on Z etc. is for the antisymmet-ˆw5x

ric indices a a PPP a . We did not decompose theˆ ˆ ˆ1 2 5Ž .indices in terms of 10D indices for 21e , because

they are not crucial for our purpose of getting they1 Ž y3r2 .expression for g dg up to OO R . The reason

Ž . w xwe need an extra g in 21a is explained in 27 .11Ž .In the limit R™` the algebra 21 recovers the

usual super Poincare algebra in Minkowski 10D, so´that we can see the link with the Green–Schwarz

w xsuperstring action 15 .We now compute the Maurer–Cartan form Ks

y1 Ž y3r2 .g dg up to OO R , following the general ex-pansion formula:

Ksgy1dgn n

! # " ! # "` 1w xs PPP d , h ,h , PPP ,h ,h ,Ý

n!ns1

22Ž .

where gsexp iX aP qu aQ 'exph. We keep atŽ .a a

Ž y1r2 . Ž y1 .most the OO R and OO R -terms, in order toelucidate the first effect of our 11D model on theGreen–Schwarz superstring action. As a simple di-mensional consideration, as well as the commutatorsthat are vanishing, we see that all the terms atŽ y1 .OO R are exhausted at the fourth commutator

Ž . Ž .ns4 in 21 .Considering all of these, we can get the expres-

sion

gy1dgsqidX aP qdu aQa a

1aq qia dug u PŽ .1 a2

1w x5qia dug u ZŽ .5 w5x'R


1 aayb dX ug QŽ .2 a aR

1 aaqb dug X QŽ .2 a aR

1cdqa dug u MŽ .2 cdR

1 1 abq yb dug u ug QŽ . Ž .2 b a6 R

1a byia b dug g u X PŽ .1 2 b aR

1 aw x5qia b dug u ug QŽ . Ž .w x5 3 5 aR1

aqia b dX uu PŽ .1 2 aR1

aby2 ia dug u X PŽ .2 a bR1 1 a

bcyi a dug u ug QŽ . Ž .2 bc a2 R

1 1aq yib a dug u uu PŽ .Ž .2 1 a24 R

1 1bc dq a a dug u ug g u PŽ . Ž .1 2 bc d2 R

1w x5 aqa a b dug u ug g g u PŽ . Ž .w x1 5 3 11 5 aR

qOO Ry3r2 . 23Ž . Ž .Ž y1r2 .All of the OO R or higher-order terms will

disappear in the limit R™`, and we are left withthe standard factor in the Green–Schwarz s-modelformulation:

1a a aKf i dX q a ug du P qdu Q . 24Ž .1 a až /2

For deriving the Type IIA Green–Schwarz actionfrom ours, the relevant traces of products of genera-

Ž < .tors in the adjoint representation of OSp 32 1 areŽ . Ž . Ž .STr PP , STr PQ and STr QQ , while for the

Ž .WZNW-term the relevant traces are STr PPP ,Ž . Ž . Ž .STr PPQ , STr QQP and STr QQQ . Among

these, the non-vanishing ones are normalized as

STr P P sh ,Ž .a b ab

iSTr P Q Q s a g g . 25Ž . Ž .Ž . aba a b 1 11 a2

In the process of getting the action for 10D Typew xIIA Green–Schwarz superstring 15 , the usual

w xfermionic k-symmetry 24 , in particular how tounderstand it in our formulation, plays a role of animportant guiding principle. We regard the k-sym-metry as a restricted gauge transformation given likeŽ . Ž .6 or 7 , under the condition that the gauge field isvanishing: As0, which is stronger than a pure-gauge condition, while acting only on K , but not on

Ž .A, as opposed to Eq. 6 . Since K is a pure-gauge,Ž .its form is to be the same as 23 , after the limit

R™`. We can identify this K with the ‘pull-back’w x Aused in the Green–Schwarz s-model 15 : K si

P A ' E Z M E A, where we identify the indexŽ .i i M

index a for Q with that of the fermionic coordi-a

nates in superspace. In fact, as a simple algebrareveals, the pull-back P A satisfies the pure-gaugei

condition dPqP 2 s0 on a flat 10D background,so that this identification is consistent.

Ž .Thus the k-symmetry acts on K similarly to 6 ,but leaving A intact:

w xd Ksd d h q K ,d h , d As0 , 26Ž . Ž .k k k k

where d h'gy1d g. This also maintains the pure-k k

gauge condition of K. To be more specific, we have

d hsgy1d gk k

m a m as d X E q d u E PŽ . Ž .k m k m a

q d u m E aQ . 27Ž .Ž .k m a

As usual, the P -term vanishes under the conditionaim md X s a d u g g u , while the Q -termŽ .k 1 k 11 a2

corresponds to the usual k-parameter, as a shift ofthe fermionic coordinates u m. In this sense, d h hask

only the fermionic component: d hs d h aQ .Ž .k k a

In order to get an explicit link with the Green–Schwarz superstring action, we take the limit R™`,

Ž < .and restrict the original OSp 32 1 group down to thesuper Poincare group. This corresponds to a trunca-´tion of components in the STr -operation. We use thesymbol STr X for such a truncated trace, in which thegenerators in fermionic contraction K aK are ex-i j a

Ž < .cluded. By this restriction, the original OSp 32 1gauge invariance is lost, but the action S has theGS

fermionic k-symmetry instead as a reminiscent of


Ž < .the original OSp 32 1 gauge invariance. ApplyingŽ .this prescription to our action S in 17 for ds2

Ž .with a s0 for d/2 , we getd

1X2 i j'S s d z yg g STr K K qbŽ .HGS i j 2

2 2S

12 i j a's d z yg g K K qb . 28Ž .H i ja 2

2 2S

We can replace this sort of rather ‘artificial’restriction of STr by considering some rescaling of

Ž .the algebra 22 after the R™`. To be specific, ifwe rescale P and Q bya a

X X'P sj P , Q s j Q , 29Ž .a a a a

putting also an additional rescaling factor jy2 inŽ .front of our action 4 , and take the limit j™`,

Ž .then we get exactly the same action as 28 :lim lim SsS for ds2. Namely, the re-j™` R™` GS

stricted supertrace STr X can be understood as such alimiting process. Of course we can just define thesupergroup G to be super Poincare and the invariant´tensor in the superalgebra to be given by the tracejust considered. At any rate, our point in obtaining it

Ž < .from the OSp 32 1 model is to show that it appearsas the result of a limiting process, not that the formeris equivalent to the later.

The k-invariance of S can be investigated moreGS

explicitly: First, we note the gauge transformation ofb is2

d b syI 1 0, K ,l sq STr K 2l . 30Ž . Ž . Ž .G 2 2

Second, since our k-transformation is a restrictedŽ .gauge transformation of K , we can read from 30

that

d b sdz i ndz jk 2

=ba aSTr K P K Q , d h QŽ .ž /i a j a k b

bi j 2 a asyia e d z K K g g d h , 31Ž . Ž . Ž .ab1 i j 11 a k

and similarly,

1i j a'd yg g K Kk i ja2

bi j a a's yg g K ia K g g d hŽ . Ž .abi 1 j 11 a k

1i j a'q d yg g K KŽ .k i ja2

bi j a a's ia yg g K K g g d hŽ . Ž .ab1 i j 11 a k

1i j a'q d yg g K K , 32Ž .Ž .k i ja2

Ž .where we have used 25 . As is easily seen now, theŽ . Ž .addition of 31 to 32 yields d S s0. In fact, ifk GS

w xwe set up the k-transformation 28 asaaad hs iP g k Q ' i Pu k Q ,Ž . Ž .k y a q a y q a

i a bd V s4a P g k '4a P g k ,Ž . Ž .k q 1 q 11 qb 1 q 11 qa

i 'd V s0 , d yg sd Vs0 , 33Ž .k y k k

where q,y are local Lorentz indices in the light-cone frame, V i are the zweibeins: g i j 'V jV j

" q yqV iV j, and we identified K A 'P A. Now wey q i i

easily see that

1i j a'd yg g P Pk i ja2

asy2 a V P g k P P ,Ž .1 q 11 q y ya

ad b sq2 a V P g k P P , 34Ž .Ž .k 2 1 q 11 q y ya

i.e., d S s0 for the total action in the Type IIAk GSw xGreen–Schwarz superstring 15,28 . Notice that the

peculiar STr X-operation excludes the derivative termd d h aQ as usual in the Green–Schwarz formula-Ž .k a

w xtion 15 .

4. Concluding remarks

In this Letter, we have proposed a new action forwextended objects on Chern–Simons supergravity 4–

x6,9 , and discussed its possible link with 10D Typew xIIA Green–Schwarz superstring theory 15 . Our ac-

tion describe in general a system of interacting braneswith or without boundaries with a gauge symmetryof supersymmetric type and generally covariant. Theequations of motion of the system can be interpretedas if the different branes carry distributional chargesin their world-volumes.

As for the quantization of this system, we canquantize it formally in the path integral approachwhere we should in principle sum over all possibleconfigurations and topologies of this system. How-ever such an ungrateful task might not be required,

Ž .as we may expect on the basis of the presumed


absence of counterterms8 with the required symme-tries that our action is in fact the effective actionalready, then all what we need is to consider abackground field perturbative expansion around a

Žsolution of the classical equations of motion the. 9vacuum .

We may also get the link with Type IIB Green–w xSchwarz superstring 15 . To this end, we have to

consider a 11D slab with two 10D boundaries forwhich we identify the gauge parameters with the10D coordinates. We chose one half of the 32 com-ponents of the spinor to be zero by imposing a Weylcondion with respect to the g so the spinor is a11

10D Majorana–Weyl spinor. Both 10D boundariesare going to be ‘identified’ in the sense that we aregoing to assume that the gauge parameters X a corre-spond to the same 10D space, however that identifi-cation can be done in two ways, related by parity.Those ways are equivalent regarding the X a, but thespinors in each boundary will have different chirali-ties. To be more specific, suppose we have twoparallel boundaries. The 11D Majorana spinor has 32components on the first boundary with the compo-

Ž . Ž .nents splitting as 32 s 16 ,16 , where andL R L R

denote the chiralities in 10D. Similarly on the secondŽ X. Ž X X .boundary, a Majorana spinor has 32 s 16 ,16 .L R

Among these four kinds of Majorana–Weyl spinorson the two 10D boundaries ‘identified’, we cantruncate either the combination 16 , 16X or 16 , 16X .R L R R

We have in each case Type IIA and Type IIBsuperstring actions, respectively.

Notice that even though the reduced action lookslike an action of a string embedded into flat super-space, our conceptual frame is quite different, be-cause the Green–Schwarz coordinates are gauge pa-rameters in our case.

We believe that heterotic strings could be alsoobtained, if we take the y’s to be nonzero andcombine u ’s and y’s to get commuting objects to

Ž . Ž .pass from Sp 32 to SO 32 , and alternatively from

8 Possibly assuming also additivity under Stieffel sum and theabove mentioned quantization of the a ’s.

9 A candidate to a vacuum that seems interesting is AdS4 =S7

with the spatial sections of the branes contained into the S7 part,except maybe for the 8-brane which may be absent or have four ofits spatial dimensions in each sector. It may be that the interactionbetween the branes is responsible for the smallness of the S7 part.

Ž . Ž . Ž . Ž .Sp 16 =Sp 16 to SO 16 =SO 16 which com-bined with a similar ‘bosonization’ of halves of Qa

gives E =E . Of course we could get the gauge8 8Ž < .groups just by adding them to the OSp 32 1 super-

group, because we have no constraints so far onwhich group we use, but that would not be so terseconceptually. Type I strings do presumably corre-sponds to the the degenerate case of a flattenedmembrane.

We could also consider adding a non pure-gaugecontribution to the potential and dimensionally re-duce it by assuming the gauge potential is indepen-dent of the membrane and background coordinatesacross the slab. Then proceeding in the same way as

w xin Ref. 4 in reducing the 3D Chern–Simons actionŽto a 2D topological action, we would have for some.choices of our symmetric supertrace only an addi-

tional term in the string action that looks like thedilaton times the curvature scalar of the stringworld-sheet, where the dilaton is essentially theeleventh component of the vielbein and the 2D cur-vature tensor is the one obtained pull-back of the11D one. This seems to provide a heuristic link tothe relationship 11D size-dilaton-coupling strength.Notice that even though we can order the contribu-tions of different string configurations to the pathintegral by the power of that coupling strength andrelate that to the genus of the string world-sheet, sothat it looks like we have a first quantized theory, wewould actually have a second quantized theory, andour classical strings are already first quantized ob-jects.

w xChern–Simons supergravity theories 5,6 have sofar developed with no explicit direct link with the

w xconventional supergravity or superstring theories 15 ,in the sense that the translation operators in theformer P in supergroups do not really generatea

‘infinite’ dimensional diffeomorphism. In otherwords, all the generators in supergroups generateonly ‘internal’ symmetries, separated from thespace-time symmetries. It has been only in a certainspeculative limit that the diffeomorphism is sup-posed to be generated or identified with the Pa

w xoperators in Chern–Simons supergravity 9 . An im-portant step in that direction has been taken recently

w xby M. Banados 29 , who added a small cosmologi-˜cal constant term to the Chern–Simons supergravityaction in 5D in order to make contact with standard


supergravity in 5D. It seems plausible that the ex-tended objects of our model may provide such anadditional cosmological constant without breakingany symmetries. Considering also the speculation

w xfrom different grounds in Refs. 6,9,29 , the 11D partof our model might have as a low energy truncationof 11D supergravity. As a supporting evidence forthis, we have provided a more solid link with 10D

w xsuperstring 15 , by taking an explicit limit to get theType IIA Green–Schwarz action in flat superspace,with supersymmetry introduced as a part of a super-group.

In our formulation, we did not introduce the targetsupergravity. The reason for this is that within ourconceptual framework, a natural extension to a moregeneral background is not a string mapped into

Ž .curved superspace, but the action given by 4 orŽ .17 . Even though this looks like a major drawbacksin our formulation at first sight, we emphasize thatthis situation is similar to the conventional M-theoryformulation, in which the particular large N-limit inthe 1D matrix model is believed to reproduce the

w xcurved supergravity background in 11D 30 . In otherwords, it is natural to expect that our microscopicmodel of Chern–Simons supergravity may well re-produce the ‘ordinary’ curved supergravity back-ground for the target space in a certain limit forsuperstring theory.

It is important to notice that our model is not anddoes not in any obvious way reduce to supermem-

w x Ž .brane theory 31 , which has closed 2 q 1 -dimensional objects with a quadratic kinetic term aswell as a WZNW-term in the world-volume. How-

w xever, quoting from the first reference of 12 refer-w xring to M-theory 10–12 , ‘‘ . . . it is not obÕious that

the presence of a membrane in Ds11 implies theexistence of a supermembrane theory, . . . but it isimportant to appreciate that, howeÕer things turnout, the major premise of M-theory, for which thecircumstantial eÕidence is now oÕerwhelming, is thatthere exist some consistent supersymmetric quantumtheory in Ds11 containing membranes and 5-branes,10 with Ds11 supergraÕity as its effectiÕefield theory.’’

10 The 5-branes are solitonic objects, while the membranes arefundamental objects

Considering all the previous points, it may notseem so implausible to conjecture that a model of the

Ž < .kind presented here for the supergroup OSp 32 1wand 11D may actually correspond to M-theory 10–

x12 , understood as the continuum limit of the under-lying pregeometric theory we have mentioned be-fore, which could be called ‘covariant matrix theory’.

We also stress that our formulation will provideus with a link between Chern–Simons supergravityw x5,6,9,29 and the conventional superstringrsuper-

w xgravity or M-theory 10–12 , in the sense that theGreen–Schwarz superstring action for the latter isrecovered in an explicit limit from the former, eventhough for the present time we can show that thisprocess works only for flat backgrounds.

Clearly much more development is expected, es-pecially concerning any constraint in the choice ofthe supergroup G and the dimensionality of thetheory and the extended objects coming from consis-tency considerations. It might be useful to try totranslate the reasoning leading to such constraints insuperstring theory into the formulation presentedhere. It might also be that these questions have anatural answer within the context of a future pregeo-metric theory having a model of the kind proposedhere as its effective theory.

Acknowledgements

We are grateful to S.J. Gates, Jr., for stimulatingdiscussions, and reading the manuscript. We are alsoindebted to the referee who helped to improve anearlier version of the paper.

References

w x Ž .1 A. Achucarro, P.K. Townsend, Phys. Lett. B 180 1986 89.w x Ž . Ž .2 E. Witten, Nucl. Phys. B 311 1988 46; Phys. B 323 1989

113.w x Ž .3 H. Nishino, S.J. Gates Jr., Int. J. Mod. Phys. A 8 1993

3371.w x Ž .4 A.H. Chamseddine, Phys. Lett. B 233 1989 291; Nucl.

Ž .Phys. B 346 1990 213.w x Ž .5 M. Banados, R. Troncoso, J. Zanelli, Phys. Rev. D 54 1996˜

2605.w x Ž .6 R. Troncoso, J. Zanelli, Phys. Rev. D 58 1998 101703;

Higher Dimensional Gravity and Local Anti-de Sitter Sym-metry, hep-thr9907109.

w x Ž .7 M.B. Green, Phys. Lett. B 223 1989 157.


w x Ž .8 G. Moore, N. Seiberg, Phys. Lett. B 220 1989 422.w x Ž .9 P. Horava, Phys. Rev. D 59 1999 046004.ˇ

w x Ž .10 C. Hull, P.K. Townsend, Nucl. Phys. B 348 1995 109.w x Ž .11 E. Witten, Nucl. Phys. B 443 1995 85.w x12 P.K. Townsend, Four Lectures on M-Theory, in: Proc. of

ICTP Summer School on High Energy Physics and Cosmol-ogy, Trieste, June 1996, hep-thr9612121; M-theory from itsSuperalgebra, hep-thr9712004.

w x Ž .13 R.P. Zaikov, Phys. Lett. B 211 1988 281; Phys. Lett. B 263Ž . Ž .1991 209; Phys. Lett. B 266 1991 303.

w x Ž .14 J.A. Dixon, M.J. Duff, E. Sezgin, Phys. Lett. B 279 1992265.

w x15 M.B. Green, J.H. Schwarz, E. Witten, Superstring Theory,Vols. 1r2, Cambridge Univ. Press, 1987.

w x Ž .16 B. Zumino, in: B.S. De Witt, R. Stora Eds. , Relativity,Groups and Topology II, North-Holland, Amsterdam, 1984.

w x Ž .17 L. Alvarez-Gaume, P. Ginsparg, Ann. of Phys. 161 1985´423.

w x18 E. Bergshoeff, F. Delduc, E. Sokatchev, Phys. Lett. B 262Ž .1991 444.

w x Ž .19 C. Hull, E. Witten, Phys. Lett. B 160 1985 398.w x Ž .20 M. Duff, B. Nilsson, C. Pope, Phys. Lett. B 163 1985 343.w x21 A.A. Tseytlin, Born–Infeld action, supersymmetry and string

theory, hep-thr9908105.

w x22 H. Nishino, Non-Linear Realization of / -Extended Super-0

symmetry, hep-thr0002029.w x Ž .23 I. Bars, Phys. Rev. D 55 1997 2373.w x Ž .24 M. Green, J.H. Schwarz, Phys. Lett. B 136 1984 367; W.

Ž .Siegel, Phys. Lett. B 128 1983 397.w x Ž .25 R. D’Auria, P. Fre, Nucl. Phys. B 201 1982 101; Nucl.´

Ž . Ž .Phys. B 206 1982 496 E ; T. Hurth, P. van Nieuwen-Ž .huizen, A. Waldron, C. Preitschopf, Phys. Rev. D 55 1997

7593.w x26 See, e.g., J.W. Holten, A.V. Proyen, J. Math. Phys. 15

Ž .1982 3763.w x )27 H. Nishino, S.J. Gates Jr., The Report, hep-thr9908136.w x28 S. Bellucci, S.J. Gates Jr., B. Radak, S. Vashakidze, Mod.

Ž .Phys. Lett. A 4 1989 1985.w x29 M. Banados, Gravitons and Gauge Fields in 5D Chern–˜

Simons Supergravity, hep-thr9911150.w x30 T. Banks, W. Fischler, S.H. Shenker, L. Susskind, Phys.

Ž .Rev. D 55 1997 5112; For reviews, see, e.g., T. Banks,TASI Lecture Note on Matrix Theory, hep-thr9911068, withreferences therein.

w x31 E. Bergshoeff, E. Sezgin, P.K. Townsend, Ann. of Phys. 185Ž .1988 330.

fundamental extended objects for chern–simons supergravity

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